May 9, 2012 Kateˇ rina Staˇ nkov´ a Helisov´ a Estimating parameters Estimating parameters in spatio- temporal Quermass- in spatio-temporal interaction process Quermass-interaction process Kateˇ rina Staˇ nkov´ a Helisov´ a Czech Technical University in Prague helisova@math.feld.cvut.cz ◭◭ joint work with Viktor Beneˇ s and Mark´ eta Zikmundov´ a ◮◮ Charles University in Prague ◭ ◮ 9th May 2012 Back Close
May 9, 2012 Outline Kateˇ rina Staˇ nkov´ a Helisov´ a 1. Quermass-interaction process and its extension Estimating parameters in spatio- temporal Quermass- interaction 2. Simulation process 3. Spatio-temporal Quermass-interaction process 4. Maximum likelihood method using MCMC 5. Particle filtering ◭◭ 6. Particle MCMC ◮◮ ◭ ◮ Back Close
May 9, 2012 Outline Kateˇ rina Staˇ nkov´ a Helisov´ a 1. Quermass-interaction process and its extension Estimating parameters in spatio- temporal Quermass- interaction 2. Simulation process 3. Spatio-temporal Quermass-interaction process 4. Maximum likelihood method using MCMC 5. Particle filtering ◭◭ 6. Particle MCMC ◮◮ ◭ ◮ Back Close
May 9, 2012 Notation Kateˇ rina Staˇ nkov´ a Helisov´ a • x = b ( u, r ) ... a disc with centre in u ∈ R 2 and radius r ∈ (0 , ∞ ) Estimating parameters in spatio- temporal • x = { x 1 , . . . , x n } ... finite configuration of n discs Quermass- interaction process • U x ... the union of discs from the configuration x . • Y ... random disc Boolean model (i.e. union of discs without any interactions) with an intensity function of discs centers ρ ( u ) and probability distribution of the discs radii Q • X ... random disc process which is absolutely continuous with re- spect to the process Y ◭◭ ◮◮ ◭ ◮ Back Close
May 9, 2012 Assumptions Kateˇ rina Staˇ nkov´ a Helisov´ a • The intensity function ρ ( u ) = 1 on a bounded set S and ρ ( u ) = 0 Estimating parameters otherwise, i.e. the centers of the reference Boolean model form unit in spatio- temporal Quermass- Poisson process on S . interaction process • For any finite configuration of discs x = { x 1 , . . . , x n } , the proba- bility measure of X with respect to the probability measure of Y is given by a density f θ ( x ) =exp { θ · T ( U x ) } , c θ where ◭◭ – c θ is the normalizing constant, ◮◮ – θ is d -dimensional vector of parameters, ◭ – T = T ( U x ) is a d -dimensional vector of geometrical characteris- ◮ tics of the union U x of the discs from the configuration x . Back Close
May 9, 2012 Quermass-interaction process Kateˇ rina Staˇ nkov´ a Helisov´ a The density is of the form Estimating parameters in spatio- f θ ( x ) = 1 temporal exp { θ 1 A ( U x ) + θ 2 L ( U x ) + θ 3 χ ( U x ) } , Quermass- interaction c θ process where • A = A ( U x ) is the area, • L = L ( U x ) is the perimeter, • χ = χ ( U x ) is the Euler-Poincar´ e characteristic (the number of con- nected components minus the number of holes, i.e. χ ( U x ) = N cc ( U x ) − N h ( U x ) ) ◭◭ ◮◮ ◭ of the union U x . ◮ Back Close
May 9, 2012 Extended Quermass-interaction process Kateˇ rina Staˇ nkov´ a Helisov´ a • Møller, Helisov´ a (2008): Estimating parameters – In the density in spatio- temporal Quermass- interaction f θ ( x ) =exp { θ · T ( U x ) } process , c θ we have T = ( A, L, χ, N h , N bv , N id ) , where N bv = N bv ( U x ) is the number of boundary vertices, N id = N id ( U x ) is the number of isolated discs, of the union U x . – Theory and simulations studied. ◭◭ • Møller, Helisov´ a (2010): ◮◮ ◭ – T = ( A, L, N cc , N h ) . ◮ – Statistical analysis. Back Close
May 9, 2012 Outline Kateˇ rina Staˇ nkov´ a Helisov´ a 1. Quermass-interaction process and its extension Estimating parameters in spatio- temporal Quermass- interaction 2. Simulation process 3. Spatio-temporal Quermass-interaction process 4. Maximum likelihood method using MCMC 5. Particle filtering ◭◭ 6. Particle MCMC ◮◮ ◭ ◮ Back Close
May 9, 2012 Papangelou conditional intensity Kateˇ rina Staˇ nkov´ a Helisov´ a Definition For a finite x ⊂ S × (0 , ∞ ) and y ∈ S × (0 , ∞ ) \ x , Estimating parameters in spatio- Papangelou conditional intensity is defined as temporal Quermass- interaction process λ θ ( x , y ) = f θ ( x ∪ { y } ) /f θ ( x ) . Denoting A ( x , y ) = A ( U x ∪ y ) − A ( U x ) , L ( x , y ) = L ( U x ∪ y ) − L ( U x ) , χ ( x , y ) = χ ( U x ∪ y ) − χ ( U x ) , ◭◭ we get ◮◮ ◭ λ θ ( x , y ) = exp ( θ 1 A ( x , y ) + θ 2 L ( x , y ) + θ 3 χ ( x , y )) . ◮ Back Close
May 9, 2012 MCMC algorithm Kateˇ rina Staˇ nkov´ a Helisov´ a 1. Suppose that in iteration t , we have a configuration x t = { x 1 , . . . , x n } Estimating parameters in spatio- 2. Proposal in iteration t + 1 : temporal Quermass- interaction process (a) with probability 1/2, the proposal is x t ∪ { x n +1 } i. we accept the proposal with probability min { 1; H ( x t , x n +1 ) } and set x t + 1 = x t ∪ { x n +1 } ii. else we set x t + 1 = x t (b) else, the proposal is x t \{ x i } i. we accept the proposal with probability min { 1; 1 /H ( x t \{ x i } , x i ) } and set x t + 1 = x t \{ x i } ◭◭ ii. else x t + 1 = x t ◮◮ where H ( x t , x n +1 ) = λ θ ( x t , x n +1 ) | S | ◭ n +1 and H ( x t \{ x i } , x i ) = λ θ ( x t \{ x i } , x i ) | S | ◮ n Back Close
May 9, 2012 Outline Kateˇ rina Staˇ nkov´ a Helisov´ a 1. Quermass-interaction process and its extension Estimating parameters in spatio- temporal Quermass- interaction 2. Simulation process 3. Spatio-temporal Quermass-interaction process 4. Maximum likelihood method using MCMC 5. Particle filtering ◭◭ 6. Particle MCMC ◮◮ ◭ ◮ Back Close
May 9, 2012 Spatio-temporal Quermass-interaction Kateˇ rina Staˇ nkov´ a Helisov´ a process Estimating parameters in spatio- temporal Quermass- Zikmundov´ a, Staˇ nkov´ a Helisov´ a, Beneˇ s (2012): interaction process f θ ( k ) ( x ) = exp { θ ( k ) · T ( U x ) } , c θ ( k ) where θ ( k ) = θ ( k − 1) + η ( k ) , k = 1 , 2 . . . , T, where θ (0) fixed is given and η ( k ) are iid random vectors with Gaussian distribution N ( a, σ 2 I ) , where a ∈ R d ( d = 3 for basic Quermass- interactio process and d = 4 , 5 , 6 for extended versions), σ 2 > 0 and I ◭◭ ◮◮ is the unit matrix. ◭ ◮ Back Close
May 9, 2012 Temporal dependence Kateˇ rina Staˇ nkov´ a Helisov´ a is given within its simulation algorithm: Estimating parameters 1. Choose a fixed θ (0) in spatio- temporal Quermass- 2. Simulate parameter vectors θ ( k ) , k = 1 , 2 , . . . , T . interaction process 3. Simulate a realization x 0 (using M-H algorithm described above). 4. Simulate realizations x k , k = 1 , 2 . . . , T (M-H alg.) with the pro- posal distribution Prop k of newly added disc at time k given by Prop k = (1 − β ) · Prop ( RP ) + β · Prop ( emp ) β ∈ (0 , 1) , k − 1 , where Prop ( RP ) is a distribution of the reference process, Prop ( emp ) k − 1 is the empirical distribution obtained from the configuration x k − 1 ◭◭ and β is a chosen constant. ◮◮ Remark: Idea is that β describs the power of time dependence so that ( β × 100)% of the added ◭ discs are taken from the previous configuration and the remaining discs are simulated randomly, so ◮ the dependence is stronger when β is bigger. Back Close
May 9, 2012 Example Kateˇ rina Staˇ nkov´ a Helisov´ a Estimating parameters in spatio- temporal Quermass- interaction process ◭◭ ◮◮ A realization of the ( A, L, N cc , N h ) -interaction process in S = [0 , 10] × [0 , 10] with Q the uniform ◭ distribution on the interval [0 . 2 , 0 . 7] , θ (0) = (0 . 5 , − 0 . 25 , − 0 . 5 , 0 . 5) , a = ( − 0 . 1 , 0 . 05 , 0 . 1 , − 0 . 1) , ◮ σ 2 = 0 . 001 and β = 0 . 5 in times k = 0 , 1 , 2 (upper row) and k = 3 , ..., 10 (lower row). Back Close
May 9, 2012 Outline Kateˇ rina Staˇ nkov´ a Helisov´ a 1. Quermass-interaction process and its extension Estimating parameters in spatio- temporal Quermass- interaction 2. Simulation process 3. Spatio-temporal Quermass-interaction process 4. Maximum likelihood method using MCMC 5. Particle filtering ◭◭ 6. Particle MCMC ◮◮ ◭ ◮ Back Close
May 9, 2012 Maximum likelihood method using Kateˇ rina Staˇ nkov´ a Helisov´ a MCMC simulations (MCMC MLE) Estimating parameters in spatio- temporal • Denote f θ ( k ) ( x ) = h θ ( k ) ( x ) /c θ ( k ) (i.e. h θ ( k ) ( x ) = exp { θ ( k ) · T ( U x ) } is Quermass- interaction process the unnormalized density). • For an observation x , the log likelihood function is given by l ( θ ( k ) ) = log h θ ( k ) ( x ) − log c θ ( k ) = θ ( k ) · T ( U x ) − log c θ ( k ) . Problem: c θ ( k ) has no explicit expression. ◭◭ Solution: We maximize the likelihood ratio f θ ( k ) /f θ ( k ) 0 for a fixed vector ◮◮ θ ( k ) instead. ◭ 0 ◮ Back Close
Recommend
More recommend
